Question: All of the 4th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$31.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$20.00$ each for teachers and $$7.50$ each for students, and the group paid $$85.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 31}$ ${20x+7.5y = 85}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-14y = -124}$ ${20x+7.5y = 85}$ Add the top and bottom equations together. $ -6.5y = -39 $ $ y = \dfrac{-39}{-6.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+3.5y = 31}$ to find $x$ ${5x + 3.5}{(6)}{= 31}$ $5x+21 = 31$ $5x = 10$ $x = \dfrac{10}{5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {20x+7.5y = 85}$ and get the same answer for $x$ ${20x + 7.5}{(6)}{= 85}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.